My main objectives when teaching the AP Calculus AB course are to complete all the topics that appear in the AP Calculus Course Description, and to make sure that the topics are being taught in a way that continuously challenges the students throughout the year. The primary textbook is Calculus of a Single Variable, 7 th ed. written by Ron Larson, Robert P. Hostetler, and Bruce Edwards. Upon completion of this course, students will have developed the tools necessary to do well in subsequent mathematics courses. This will be achieved by balancing a basic understanding of the ideas and concepts, the development of particular skills and the use of technology throughout the course.
Key for Requirements Met
Label Curricular Requirement
[C2] The course teaches all topics associated with Functions, Graphs, and Limits;
Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.
[C3] The course provides students with the opportunity to work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.
[C4] The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.
[C5] The course teaches students how to use graphing calculators
[C2]
Chapter P Preparation for Calculus
(Students complete a review of these Precalculus topics over the summer)
Chapter 1 Limits and Their Properties
(3 weeks)
1.
Finding Limits Graphically and Numerically
2.
3.
Evaluating Limits Analytically
Continuity and One – Sided Limits
4.
Infinite Limits
(Activity #1- 3 Ways to Find a Limit)
Chapter 2 Differentiation
(4.5 weeks)
1.
The Tangent Line Problem
2.
Basic Differentiation Rules and Rates of Change
3.
The Product and Quotient Rules and Higher Order Derivatives
4.
The Chain Rule
5.
Implicit Differentiation
6.
Related Rates
(Activity #2 “Lollipop Lab”)
Chapter 3 Applications of Differentiation
(5 weeks)
1.
2.
Extrema on an Interval
Rolle’s Theorem and Mean Value Theorem
3.
Increasing and Decreasing Functions and the First Derivative Test

4.
Concavity and the Second Derivative Test
5.
Limits at Infinity
6.
A Summary of Curve Sketching
7.
Optimization Problems
8.
Local Linearity and Newton’s Method
Differentials 9.
(Activity #3 Find your Calculus Mate)
(Activity #4 Can Price Optimization Project)
Chapter 4 Integration
(4.5 weeks)
1.
Antiderivatives and Indefinite Integration
2.
Area
3.
Riemann Sums and Definite Integrals
4.
The Fundamental Theorem of Calculus
5.
Integration by Substitution
6.
Numerical Integration including Trapezoidal Rule with Error Calculation
(End of Semester 1)
Chapter 5 Logarithmic, Exponential, and other Transcendental Functions
(4 weeks)
1.
2.
The Natural Logarithmic Function (Differentiation)
The Natural Logarithmic Function (Integration)
3.
Inverse Functions
4.
Exponential Functions (Differentiation and Integration)
5.
Slope Fields
Differential Equations (Growth and Decay) 6.
7.
Differential Equations (Separation of Variables)
8.
Inverse Trigonometric Functions (Differentiation and Integration)
Chapter 6 Applications of Integration
(4.5 weeks)
1.
Area of a Region Between Two Curves
2.
Volume: Disk Method
3.
4.
Volume: Shell Method
Volume: With Known Cross Section
5.
Arc Length
(Activity #5 Fruit Lab)
(Review for AP Exam and AP Exam Administration 1.5 weeks)
Chapter 7 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals
(3 weeks)
1.
Integration by Parts
2.
Trigonometric Substitution
3.
Partial Fractions
4.
L’Hôpital’s Rule
5.
Improper Integrals
(Activity #6 Integration By Parts Go fish!)

(The following chapters are alternated year to year or are abbreviated as time allows.)
Chapter 8 Infinite Series
(2 weeks)
1.
Series and Convergence
2.
3.
The Integral Test and p-Series
Comparisons of Series
4.
Alternating Series
5.
The Ratio and Root Tests
6.
Taylor Polynomials/Maclaurin Polynomials and Approximations
Chapter 9 Conics, Parametric Equations and Polar Coordinates
(2 weeks)
1.
Review of Parametric Equations and Graphs
2.
Parametric Equations and Calculus
3.
Polar Coordinates and Polar Graphs
4.
Area and Arc Length in Polar Coordinates
5.
Polar Equations and Conics
(Performance Based Assessment and Final Exam 1.5 weeks)
On the first day of school, I begin with limits right away. From limits, after the limit activity which is mentioned in the student activity section, a limit definition is introduced and compared to what it means to find an average rate of change.
Derivative rules follow with explanations for each one as well as the derivations and proofs verifying these rules. Applications of each topic are always incorporated when possible and were appropriate.
Throughout the course, students work together on a regular basis, both formally and informally. At times groups will even form without my having to tell the students to form those groups. Because my room has lab tables instead of regular desks, students will start working in groups even when going over homework or discussing material for a new section.
When new topics are being presented the lesson format is designed to be as student centered as possible. Through discovery lessons, group discussions, think –pair share work, or explorations, students take ownership of the new material because they are involved with the informative process.
Topics are presented through a multi-faceted approach whenever possible. Graphs, numerical analysis, analytical and algebraic analysis, and verbal explanation are all used when presenting topics, and assessing the student’s knowledge of these topics.
[C5], [C3]
Technology is used throughout the course on a regular basis. All students in the class are issued TI-83+ calculators at the beginning of the year and are required to have it in class every day. In my class, we use the graphing calculators daily to explore, discover, connect, and reinforce the concepts of calculus. I also use TI
SmartView on a regular basis when demonstrating different applications or techniques on these calculators. TI Connect is often used during labs in class so the students can capture graphs they have created and incorporate them into their lab reports.

Throughout the year, I assign several AP free response questions for students to work on both outside of class and during class time. Once these questions have been collected and graded. A similar question will be included on the formal assessment for each topic. Every test given to the students is created within the AP format, both calculator and non-calculator portions are given along with multiple choice and free response. As the AP Calculus test gets closer a full mock test is given in stages over three or four periods in order to prepare the students fully and help me diagnose what areas of content will give them trouble.
At the end of the first semester a project is assigned to the students individually that requires them to create their own AP free response problem given certain guidelines that have to me met. They are required to write a multi-part question using one or more topics we have discussed so far this year. An answer key and full rubric should also be included with the break down of point distribution.
Larson, Ron et al. Calculus of a Single Variable. 7 th ed. New York: Houghton Mifflin
Company, 2002.
1. 3 Ways to Find a Limit
Students work in small groups after initial instruction has been given on limits and complete a packet and present their findings. Each group is given a different type of function and their task is to demonstrate the value of its limit as x goes to some a in three different ways, graphically, numerically (through table exploration on calculator), and analytically. As a connective question, students are asked later in a journal entry if the solutions found in this assignment are related to the value of the derivative of the function at a. Reasoning must also be included. [C3]
2. Lollipop Lab
During the study of related rates, student will complete a lab which requires them to suck on Tootsie Roll Pops to determine the rate of change in their radius; they then can calculate the rate of change in the pop’s volume. Students measure the initial radius of a Pop with dental floss. They then suck on the Pop for 30 seconds, record its radius, suck on the pop for 30 seconds, etc. They then use the regression capabilities of their calculator to model the rate of change of the radius with some function of time. Students then use this rate of change to estimate the rate of change of the volume of the Pop when its radius is three fourths its original radius.
This lab, “How Many Licks” can be found in Ellen Kamischke’s book, A Watched Cup
Never Cools.
3. Find Your Calculus Mate
After the class has some familiarity with graphs of functions and their derivatives, this activity will start the next class. A Set of graphs of functions and their derivatives will be prepared and each pair will be up on two separate cards. (One graph per card) As students come in to class all the cards will be handed out and they will have to find the graph that belongs with theirs. Once they have found their graph each pair will be taped on the board and we will have a group discussion to see if the class agrees with the pairs and why they agree or not. When all the graphs have been paired to the satisfaction of the class, the equations for each function will be given and the students will have to find the derivatives when

possible. They will use the graphing calculator to confirm the correctness of the pairings. This activity can be found in Lin McMullin’s book, Teaching AP
Calculus.[C4] [C5]
4. Can Price Optimization Lab
The study of optimization has a great direct application when discussing packaging of products. Students will be asked to design an optimum can for mass production.
Students obtain a can of soda, soup, or any other cylindrical container. They measure the height and diameter of the can and determine it’s volume. They then find the radius and height that would produce the most cost – effective can that will still hold the same volume. The students will also be required to write an explanation of the mathematics involved in making their determination. Students will construct a mock can with their calculated dimensions and use it as a visual comparison to the original can during their explanation in class. The background and idea for this activity came from Calculus of a Single Variable on page 159. [C4]
5. Fruit Lab
Working in teams of two, students will pick a piece of fruit and determine the volume of the fruit using volume displacement measuring techniques. Students will then cut the fruit into quarters and use 1cm square graph paper to plot points along the outside of one of these quarters. These data points will be entered using list capabilities of the TI – 83+ calculator and they will be plotted using STAT PLOT. A function that fits the points will be calculated and used to find the volume of the fruit by revolving the curve around the x-axis. The resulting solid of revolution should represent the whole fruit pretty closely. Students will also discuss possible reasons why the volumes may have differed in the conclusion of their lab along with their findings. This Lab was developed and presented at the Rutgers University
Precalculus Conference in 2004. [C5]
6. Integration by Parts Go Fish
In order for the students to become more comfortable deciding which portion of an integral to assign to v and which portion to assign to dv, a set of “cards” are created where there are u cards, dv cards, and complete integral cards. The game involves the regular “Go Fish” rules but to but cards down they have to make a triple, the matching, u, dv and integral. Repeated use of this game puts the students more at ease when solving integrals using this technique. This game was originally developed for use within the Air Force Academy’s Calculus classes.